A sequence where every step adds the same amount. Two formulas cover everything you are asked about one: a rule for any single term, and a rule for the total of a run of them. The calculator below will work backwards from whichever values you happen to have.
The two formulas, and what the letters mean
| Symbol | Meaning |
|---|---|
| a | The first term, sometimes written a₁. |
| d | The common difference — what is added each step. Negative if the sequence falls. |
| n | How many terms, or which term you want. The first term is n = 1, not n = 0. |
| aₙ | The value of the nth term. |
| Sₙ | The total of the first n terms. |
The nth-term rule, aₙ = a + (n − 1)d, says: start at a, then take d exactly n − 1 times. It is n − 1 rather than n because getting to the first term takes no steps at all. That single off-by-one is responsible for most wrong answers on this topic.
The sum has two forms. Sₙ = n⁄2 (a + aₙ) is the one to use when you know the last term; Sₙ = n⁄2 [2a + (n − 1)d] is the same thing with aₙ substituted out, for when you do not.
Why the sum formula looks like that
Write the sum forwards, then write it again backwards underneath, and add the two lines column by column. Every column comes to the same total, a + aₙ, and there are n columns. That gives you twice the sum, so the sum itself is n⁄2 (a + aₙ).
The story attached to this is that Gauss found it as a schoolboy when told to add the numbers from 1 to 100, pairing 1 with 100, 2 with 99, and so on: fifty pairs of 101, so 5050. Whether or not the story is true, the pairing is the whole idea, and it is worth understanding rather than memorising because the derivation is a common exam question in itself.
A worked example
A sequence starts at 7 and rises by 4 each time: 7, 11, 15, 19, …
What is the 20th term?
a₂₀ = a + (n − 1)d = 7 + (20 − 1) × 4 = 7 + 19 × 4 = 7 + 76 = 83
What do the first 20 terms add up to?
S₂₀ = n⁄2 (a + aₙ) = 20⁄2 × (7 + 83) = 10 × 90 = 900
Which term is 143? Now the unknown is n, so rearrange:
143 = 7 + (n − 1) × 4 → 136 = (n − 1) × 4 → n − 1 = 34 → n = 35
Note that n came out a whole number. If it had not — say the question asked which term is 145 — the answer would be that 145 is not in this sequence at all. That is a legitimate answer and questions are sometimes set to catch it.
Where marks get lost
Using n instead of n − 1. The most common error on this topic by a distance. The 20th term takes nineteen steps, not twenty.
Getting the sign of d wrong. In a falling sequence such as 50, 46, 42, … the common difference is −4. Feeding in +4 gives a plausible-looking answer that is wrong.
Confusing aₙ with Sₙ. “The 10th term” and “the sum of the first 10 terms” are different questions. Read carefully which one is being asked for; the word “sum” or “total” is your signal.
Assuming a sequence is arithmetic when it is not. Check that the difference really is the same between every pair of consecutive terms. If each term is a fixed multiple of the one before, it is a geometric progression and these formulas do not apply.
Accepting a fractional n. n counts terms, so it has to be a positive whole number. A fractional answer means the value you were given is not a term of that sequence.
Questions students actually ask
What is the difference between an arithmetic and a geometric progression?
An arithmetic progression adds a fixed amount each step; a geometric one multiplies by a fixed amount. 3, 7, 11, 15 is arithmetic with d = 4. 3, 6, 12, 24 is geometric with a ratio of 2.
Why is it (n − 1)d and not nd?
Because reaching the first term takes no steps. The second term is one step from the start, the third is two, and the nth is n − 1. Count the gaps, not the terms.
How do I find the sum when I do not know the last term?
Use the other form, Sₙ = n⁄2 [2a + (n − 1)d]. It is the same formula with aₙ replaced by a + (n − 1)d, so it needs only a, d and n.
Can the common difference be zero?
Yes. Every term is then the same and the sum is simply n × a. It is a valid arithmetic progression, just a dull one — and it is the case where you cannot work n out from the terms, since every term looks alike.
What if the sequence is decreasing?
Nothing changes except that d is negative. The formulas handle it directly, and the sum can come out negative if the sequence runs far enough below zero.
Related formulas
The quadratic formula · Pythagoras’ theorem · Moles, mass and molar mass
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